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Meet Representations in Upper Continuous Modular Lattices James Elton Delany Iowa State University

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Meet Representations in Upper Continuous Modular Lattices James Elton Delany Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1966 Meet representations in upper continuous modular lattices James Elton Delany Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Delany, James Elton, "Meet representations in upper continuous modular lattices " (1966). Retrospective Theses and Dissertations. 2894. https://lib.dr.iastate.edu/rtd/2894 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. This dissertation has been microiihned exactly as received 66-10,417 DE LA NY, James Elton, 1941— MEET REPRESENTATIONS IN UPPER CONTINU­ OUS MODULAR LATTICES. Iowa State University of Science and Technology Ph.D., 1966 Mathematics University Microfilms, Inc.. Ann Arbor, Michigan MEET REPRESENTATIONS IN UPPER CONTINUOUS MODULAR LATTICES by James Elton Delany A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved: Signature was redacted for privacy. I _ , r Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University of Science and Technology Ames, Iowa 1966 il TABLE OP CONTENTS Page 1. INTRODUCTION 1 2. TORSION ELEMENTS, TORSION FREE ELEMENTS, AND COVERING CONDITIONS 4 3. ATOMIC LATTICES AND UPPER CONTINUITY 8 4. COMPLETE JOIN HOMOMORPHISMS 16 5. TWO HOMOMORPHISMS 27 6. MEET IRREDUCIBLE ELEMENTS 36 7. IRREDUNDANT REPRESENTATIONS 41 8. APPLICATIONS 51 9. BIBLIOGRAPHY 58 10. ACKNOWLEDGMENTS 59 1 1. INTRODUCTION The study of representations of elements of the lattice of congruence relations of an algebra as intersections of other elements of this lattice is of interest for two reasons. In the first place, such a study offers an extension of the classical representation theory for the ideals of a noetherian ring. Secondly, there is a natural correspondence between the subdirect product representations of an algebra and the meet representations of the zero element of its lattice of con­ gruence relations. The lattice of congruence relations of any algebra is compactly generated. (An element x of a lattice is compact if whenever x 4US there is a finite set FcS such that x^ijF, A complete lattice is said to be compactly generated if each element is the union of a set of compact elements.) In order to study the subdirect product representations of an algebra it often suffices to investigate meet representations in compactly generated lattices. For example, the fact that any algebra has a subdirect product representation in terms of subdirectly irreducible factor algebras may be regarded as a consequence of the fact that each element of a compactly generated lattice has a meet representation in terms of com- I pletely meet irreducible elements. (An element x of a lattice is meet irreducible if xssj^o Sg implies that either x=s^ or xaSg. It is completely meet irreducible if x=Os implies X g S.) A meet representation of an element, say x=S, is 2 irredundant if x<fl(S - s) for each seS, Although any ele­ ment of a compactly generated lattice has meet representations in terms of completely meet irreducible elements, it is not true that it must possess an irredundant one. For example, Dilworth and Crawley have characterized the compactly generated modular lattices in which each element has such a representa­ tion as those which are atomic (2). (A lattice is atomic if whenever x<y there exists x' such that x-<x'^. Here, as throughout this paper, x -<x' signifies that x is covered by x'.) Our goal is to find compactly generated modular lattices in which each element has an irredundant representation in terms of meet irreducible elements (instead of completely meet irreducible elements.) Actually, for our purposes it will suffice to consider upper continuous modular lattices. (A complete lattice is upper continuous if for each element X and each chain of elements S the relation x A Us=(Jg (xn s) holds.) Any compactly generated lattice is upper continuous. On one hand we will see that there is an analogy between the case dealing with meet irreducible elements and that in­ volving completely meet irreducible elements. This may be seen by comparing Theorem 7.4 to Theorem 7.5, the latter being a restatement of the above mentioned result of Dilworth and Crawley. In a somewhat different vein we will show that a cer­ tain type of lattice homomorphism is useful in studying ir­ redundant representations and the role played by covers in such representations. Adopting the terminology used in (3) 3 we say that a lattice homomorphism .<|>:L-+-L* is a complete join homomorphism if L and L' are complete and for each Sc=L we have *(U S)= U S € o (s). Our principal results concerning such maps are the following. If L is an upper continuous modular lattice then there is an associated upper continuous modular lattice L* which is the "largest" homomorphic image of L (under a complete join epimorphism) possessing no covers. (See Theorem 5.3) Each element of L has an irredundant repre­ sentation in teirms of meet irreducibles if and only if each element of L* has such a representation. In particular, if L* consists of one element then each element of L has an irredundant representation in terms of meet irreducible ele^ ments. The class of upper continuous modular lattices for which L* consists of one element is large enough to include both the atomic ones, in which each element has an irredun­ dant representation in terms of completely meet irreducible elements, and those satisfying the ascending chain condition, in which each element has an irredundant representation as the meet of a finite number of meet irreducible elements. 4 2. TORSION ELEMENTS, TORSION FREE ELEMENTS, AND COVERING CONDITIONS We begin by generalizing two concepts from group theory. Definition. An element t of a lattice L is a torsion element if for each a<t there is a b in L such that t^>-a. The element 0 is always a torsion element. If 1 is a torsion element, we will call L a torsion lattice. Definition. An element of L is torsion free if no element of L covers it. Here 1 is always torsion free. Let H be a subgroup of an abelian group G, and let L(G) denote the lattice of subgroups of G. H is a torsion element of L(G) if and only if H is a torsion group. H is torsion free in L(G) if and only if G/H is a torsion free group. In fact the duals of the above definitions have counterparts in the theory of abelian groups. H is dual torsion free in L(G) if and only if H is a divisible group. H is a dual torsion element of L(G) if and only if G/H is a reduced group. In general it is not the case that a lattice must possess maximal torsion elements or minimal torsion free elements. Examples are easily constructed by violating the following condition. Definition. A lattice satisfies the lower covering condition if a>-a n b implies au b>-bl The dual of this condition is termed the upper covering 5 condition. If a and b are elements of a modular lattice then the intervals [an b,a] and [b,au b] are isomorphic. Thus a modular lattice satisfies both covering conditions. Our first theorem is a direct generalization of a clas­ sical theorem of group theory. Theorem 2.1 In a complete lattice which satisfies the lower covering condition there is an element which is both the largest torsion element and the smallest torsion free element. Before proving this we will develop some lemmas. In the remainder of this section it is assumed that the lattice under discussion is complete and satisfies the lower covering condition. Lemma 2.1 If x is a torsion element of "L, , and [y, z] has no atoms, then y^. Proof. Assume x n y<x. Since x is a torsion element there is a c such that x^c>-xn y. Now xny^cny^c so that either c n y=c or cny-xny. If c n y=c we would have c=c n x^ n X -<c. Hence c n y=x n y so that c>- c n y. By the lower covering condition, cvy>-y. But cuy^uy^z, contrary to the hypothesis that [y,z] has no atoms. Thus xn y=x and y>f. Lemma 2.1 has the following three consequences. Lemma 2.2 If x is a torsion element and y is torsion free then x^. Proof: Let z«l. Then x,y, and z satisfy the hypo­ 6 thesis of Lemma 2.1. Lemma 2.3. If T is a collection of torsion elements then U T is a torsion element. Proof; Suppose y^UT and [y/UT] has no atoms. If tfiT, t;^L/T and Lemma 2.1 implies y^t. Then y^UT. Hence y=UT and UT is a torsion element. Lemma 2.4. If t is a torsion element of L and t'>-t then t' is also a torsion element. Proof; Suppose y^t' and [y,t'] has no atoms. By Lemma 2.1, y^t. But y^t since t't. Hence y=t' and t' is a torsion element.
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